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Power expression and modulus - GMAT standard approach

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Power expression and modulus - GMAT standard approach  [#permalink]

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New post 23 Jan 2010, 11:12
1
Question : \(a^b^c\)

How one will interpret this ?\((a^b)^c\) or \((a)^(b^c)\)

Second :
\((81)^1/4\) ; It is equivalent to 3 and or -3 in GMAT ? I am aware that \(\sqrt{(x^2)}\) is always positive value.
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Re: Power expression and modulus - GMAT standard approach  [#permalink]

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New post 23 Jan 2010, 12:43
GMATMadeeasy wrote:
Question : \(a^b^c\)

How one will interpret this ?\((a^b)^c\) or \((a)^(b^c)\)

Second :
\((81)^1/4\) ; It is equivalent to 3 and or -3 in GMAT ? I am aware that \(\sqrt{(x^2)}\) is always positive value.


You would interpret \(a^b^c\) as \(a^(b^c)\) - the rule is to work 'top-down'. To me, it would feel 'funny' to evaluation a^b before evaluation b^c.

The second one confuses me:
\(\sqrt{(x^2)}\) is x always positive?
Take \(x=-2\) for example:
\((-2)^2= 4\)
\(\sqrt{4}=+2, -2\), so x could have been either positive or negative.

Hence, \((81)^1/4\) is +3 or -3, but you cannot tell from the equation.
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Re: Power expression and modulus - GMAT standard approach  [#permalink]

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New post 23 Jan 2010, 12:52
andrewcs wrote:
GMATMadeeasy wrote:
Question : \(a^b^c\)

How one will interpret this ?\((a^b)^c\) or \((a)^(b^c)\)

Second :
\((81)^1/4\) ; It is equivalent to 3 and or -3 in GMAT ? I am aware that \(\sqrt{(x^2)}\) is always positive value.


You would interpret \(a^b^c\) as \(a^(b^c)\) - the rule is to work 'top-down'. To me, it would feel 'funny' to evaluation a^b before evaluation b^c.

The second one confuses me:
\(\sqrt{(x^2)}\) is x always positive?
Take \(x=-2\) for example:
\((-2)^2= 4\)
\(\sqrt{4}=+2, -2\), so x could have been either positive or negative.

Hence, \((81)^1/4\) is +3 or -3, but you cannot tell from the equation.


I concur with both the points you mentioned. However, for first point , that is top down approach of exponent, I will post a good question in a while to solve. For second point, refer to GMAt Official guide , maths theory section ,it explicitly indicates that \(\sqrt{x^2}=|x|\) . I extended the definition further from x^2 to x^4.

I do not know what is the final answer .
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Re: Power expression and modulus - GMAT standard approach  [#permalink]

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New post 24 Jan 2010, 01:15
2
GMATMadeeasy wrote:
Question : \(a^b^c\)

How one will interpret this ?\((a^b)^c\) or \((a)^(b^c)\)

Second :
\((81)^1/4\) ; It is equivalent to 3 and or -3 in GMAT ? I am aware that \(\sqrt{(x^2)}\) is always positive value.


First question: \(a^m^n=a^{(m^n)}\) and not \((a^m)^n\), for instance \(3^{3^3}=3^{(3^3)}=3^{27}\).

Second question: \(\sqrt[_{even}]{x}\geq0\) - Even roots have only a positive value on the GMAT. (well if x=0 then it will obviously be 0). So \(\sqrt[4]{81}=3\), not +3 and -3.

Also general rule: \(\sqrt{x^2}=|x|\).

When we see the equation of a type: \(y=\sqrt{x^2}\) then \(y=|x|\), which means that \(y\) can not be negative but \(x\) can.

\(y\)can not be negative as \(y=\sqrt{some expression}\), and even root from the expression (some value) is never negative (as for GMAT we are dealing only with real numbers).

When the GMAT provides the square root sign for an even root, such as a square root, then the only accepted answer is the positive root.

That is, \(\sqrt{16} = 4\), NOT +4 or -4. In contrast, the equation \(x^2 = 16\) has TWO solutions, \(x=+4\)and \(x=-4.\)

On the other hand odd roots will have the same sign as the base of the root. For example, \(\sqrt[3]{64}=4\) and \(\sqrt[3]{-27}=-3\).

Hope it's clear.
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Re: Power expression and modulus - GMAT standard approach  [#permalink]

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New post 26 Jan 2010, 11:12
1
Let's try to solve this based on above.

What is the value of x ?

1 \sqrt{x^4} = 9
2 The average of x^2,6x and 3 is -2


P.S. First atement is square root of (x^4)
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Re: Power expression and modulus - GMAT standard approach  [#permalink]

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New post 26 Jan 2010, 11:20
GMATMadeeasy wrote:
Let's try to solve this based on above.

What is the value of x ?

1 \sqrt{x^4} = 9
2 The average of x^2,6x and 3 is -2


P.S. First atement is square root of (x^4)


(1) I would think this gives x=+3 or x=-3 as possible solutions, so not sufficient.
(2) (x^2 + 6x + 3)/3 =-2
x^2 + 6x +3 = -6
x^2 + 6x + 9 = 0
x=-3

So (2) alone, B?
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Re: Power expression and modulus - GMAT standard approach  [#permalink]

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New post 30 Jan 2010, 16:19
B;

agree with andrewcs... A doesnt give unique value, B does...
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Re: Power expression and modulus - GMAT standard approach  [#permalink]

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Re: Power expression and modulus - GMAT standard approach   [#permalink] 29 Jan 2018, 00:17
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